Modeling the Dynamics of Deformable Objects Based on Volumetric Patches of Free Forms

Abstract

A method for solving nonlinear implicit numerical integration problems based on volumetric patches of free forms for modeling deformable objects with high-speed dynamics is proposed. This method improves the accuracy, consistency and controllability of deformation modeling and animation. The method is characterized by speed, accuracy, stability and uses optimization with region decomposition. The method is well suited for modeling deformable bodies with a large time step, in a wide range of deformation dynamics. We propose decomposed optimization, an optimization method based on free-form patches with region decomposition to minimize incremental potentials at each time step. The method uses quadratic matrix decomposition to combine non-overlapping subregions. The Hessian is evaluated once at the beginning of the time step. The advantages of the method are as follows. Geometric primitives and their mathematical models are proposed, which allow the reasonable application of these primitives to solve problems of volume-oriented modeling. Such requirements are met by volumetric patches of free forms based on analytical perturbation functions relative to the base triangles. The decomposed Hessian is constructed for each subregion and calculated using a set of vertices taken from a complete non-decomposed grid. The weights add the missing second-order Hessian data to the vertices of the subregions from the neighbors along the decomposition boundaries. Thanks to this, the descent is performed along the grid coordinates. There is no need to add gradients. During the descent, the gradient is determined. The Hessians under the region are calculated and factorized in parallel once per time step. They are used as an initializer at each iteration. Then the results are mixed together. This ensures stable and continuous high-quality modeling. An automated and reliable optimization method adapted for modeling nonlinear materials, high-speed dynamics and large deformations is proposed.